Sir I am from India Do you know about JEE Advanced exam of India
@Marlow9986 күн бұрын
if f(f(y))=y and f is surjective does that also mean f(y)=y?
@littlefermat6 күн бұрын
@@Marlow998 If f(f(y)) =y then f is already surjective as f takes all values. Of course that doesn't mean f(y) =y. Take f(y) =1-y as a counter example
@Marlow9986 күн бұрын
@@littlefermat Alright thanks for the quick reply also when we assume a number alpha such that f(alpha)= smth can we assume an another numbeer beta f(beta)= smth2 ?
@NovierJohn12 күн бұрын
Where are you from?
@kaiserquasar317814 күн бұрын
Oh dear Lord, your channel is EXACTLY what I needed to find. Thank you so much for this, I'll be bingingnthis series and taking extensive notes prepping for math olympiads. Hope to see y'all at the IMO!
@Sparkles.08Күн бұрын
May I ask what country you are from?
@kaiserquasar3178Күн бұрын
@@Sparkles.08 yeah sure but for what purpose? You look like a bot if anything tbf
@Sparkles.08Күн бұрын
@@kaiserquasar3178 wow, hold up. Just wanna be friends that's it. You're preparing for an IMO if I'm not mistaken and I'm taking part in a math competition too.... that's it, I'm no harm.
@kaiserquasar3178Күн бұрын
@@Sparkles.08 Fr? Well then, I'm from Lithuania. You?
@Sparkles.0817 сағат бұрын
@@kaiserquasar3178 I'm from Namibia
@strigibird983214 күн бұрын
In the step where you conclude that 1=f(f(y))+f(y). Would it be possible to make a substitution such that u=f(y)? Which would make the equation 1=f(u)+u. This can then be rearranged to become f(u)=1-u.
@Ivnkkk24018 күн бұрын
How can we know that this is the only solution?
@Samuel-zs9gw24 күн бұрын
Hello sir, please do you offer 1 to 1 online lessons ?
Is there another word for the circle method? I have tried to search for it on Google but I didn’t get any results
@littlefermat26 күн бұрын
@@user-bi3oc2jt4t Nah, I just came up with the word as it is literally like a circle!
@user-bi3oc2jt4t26 күн бұрын
@@littlefermat But where did you learn about it? I want to learn more about this tool
@user-bi3oc2jt4t27 күн бұрын
THANK YOU!!! All the other videos on here only talked about how to prove a function is surjective, and not how to use it’s properties to solve problems. This really helped me
@littlefermat27 күн бұрын
@@user-bi3oc2jt4t glad you liked the video, the whole playlist is discussing techniques to solve functional equations.
@user-bi3oc2jt4t27 күн бұрын
@@littlefermatAwesome
@fahrenheitwastaken27 күн бұрын
Thank you!
@littlefermat27 күн бұрын
@@fahrenheitwastaken you are welcome!
@krstev2927 күн бұрын
Very fun and easy problem!
@erapesmobile421629 күн бұрын
thank you!
@littlefermat27 күн бұрын
@@erapesmobile4216 you are welcome!
@justnothingexciting667329 күн бұрын
Great🎉
@user-jm6rm2xn3z29 күн бұрын
hi sir please can you solve this functional equation i try for more times but i can't solve it f(xf(y)-f(x))=2f(x)+xy
@SyibeАй бұрын
Hey sir, I was wondering what I should do in order to prepare for math Olympiad. Im currently year 10 (gcse) and can u please recommend a few textbooks for the preparation.
@ryanstaal3233Ай бұрын
Solution 3: Let H be the orthocentre of ABC. Then HBX=HBA=HCA=HCX, so HBCX concyclic. This gives BXC=BHC=180-BAC=180-BZC=BYC. Also XCB=YCB and XB=YB, so triangles XCB and YCB are congruent. That gives BC perpendicular to XY, so BC perpendicular to ZX. Combining this with BXC=180-BZC implies that X is orthocenter of BCZ, so BZ perpendicular to CX
@ryanstaal3233Ай бұрын
After BXHC concyclic we can also prove BC perpendicular to XY by using that reflecting (BHC) in BC gives (BAC), so reflection X’ of X in BC is the point in (ABC) with BX’=BX=BA which is Y. So XY perpendicular to BC
@iMíccoli29 күн бұрын
Beautiful solution
@ryanstaal3233Ай бұрын
Another solution: B is on perp bisector of AY, so on angle bisector of AZY. So B is on angle bisector of AZX and on perp bisector of AX. But obviously ABXZ aren’t concyclic since X!=C. This means that the angle bisector of AZX is the same line as the perp bisector of AX. So Z is on perp bisector of AX. Since B is also onnthat perp bisector we have BZ perpendicular to AC
@ryanstaal3233Ай бұрын
Solved in 1 minute, solution: angle chase BZX+ZXA=BZY+180-AXY=BAY+ABY/2=90, so BZ is perpendicular to AC
@iMíccoliАй бұрын
Well I came up with a difference solution using both "bad points". So let D be the intersection of |AY| and |BZ|. Notice that since |BY|=|BA| we have that B is the mid point of the arc AY so BZ is the angle bisector of [angle AZY] and used this with the fact that ABCZY is cyclic we'll have [angle AZB]=[angle YZB]=[angle BYA]=[angle BAY]. Now notice that ∆BYD~∆BZY because [angle BYD]=[angle BZY] and they have a common angle at B therefore [angle BDY]=[angle BYZ] but the ∆BXY is isosceles so [angle BYZ]=[angle BYX]=[angle BXY]=[angle BDY] so the quadrilateral BYXD is cylic thefore [angle DYX]=[angle XBD]. With this we have [angle AYZ]=[angle ABZ]=[angle XBD] so BZ is the angle bisector of [angle XBA ] and since ∆BXA is isosceles it is also it's perpendicular bisector so that finishes the problem.
@littlefermatАй бұрын
Great job! To be honest, this problem is not that hard, that whatever angle you pick you can find a way to handle it. The strategy of reducing the number of bad points however becomes very important in much harder problems. I will upload a video explaining more with a much more suitable example.
@iMíccoliАй бұрын
@@littlefermatlooking forward to it.
@iMíccoliАй бұрын
@@littlefermatthanks
@joseluishablutzelaceijas928Ай бұрын
Thanks for the video. Here a solution involving the bad points X and Y: As |AB| = |BY|, one has that [angle AZB] = [angle ACB] = [angle BZY] = [angle BCY]. In the circle with center at B and passing through A, X and Y, one has that [angle AXY] + [angle ABY]/2 = 180°, which means that [angle AXZ] = [angle ABY]/2. As the points B and Z lie on opposite sides of AY, one has that [angle ABY] + [angle AZY] = 180° = 2*[angle AXZ] + 2*[angle BZY], i.e. [angle AXZ] + [angle BZY] = 90°, which means that AC and BZ are perpendicular.
@littlefermatАй бұрын
Nice! Indeed there are many ways to tackle this problem regardless of considering angles with bad points. However, this strategy is affective when handling angles in harder problems. I will try to post a video where I illustrate the whole principle with a really complicated angle chasing problem.
@iMíccoliАй бұрын
Nice solution
@iMíccoliАй бұрын
@@littlefermatPlease do it :).
@JohnMagar069Ай бұрын
Lets go!!!
@shafikbara440Ай бұрын
I am going to this year's IMO any advices for the contest?
@bata3258Ай бұрын
congrats mate
@littlefermatАй бұрын
Good luck my friend. Make sure to relax and have fun and don't forget to check the marking scheme with your leader in case you have written something valuable on draft during the competition. Also make sure you spend two weeks revising before you go there. Good luck again!
@extreme4180Ай бұрын
this was fairly easy problem
@littlefermatАй бұрын
Indeed
@bata3258Ай бұрын
Its a good day when little fermat uploads
@Samuel-zs9gwАй бұрын
I found the answer graphically in less than 1 minutes😭 1) the function is odd and therefore its graph is symmetrical about the origin 2) for all values of x, when we increase X by 1 f(x) is also increased by 1, therefore it had to be a line of gradient/slope 1, only the graph of y=X could hold 3) just checked if f(x)=X was true for the last one, And bam!, I know it just worked for that particular case but at that instant of my life, I felt so smart😭😭 It gives me hope Thanks so much for your videos ❤️!
@hmkl6813Ай бұрын
We had this as P2 last year during nationals with the word KLAUSUR(German for exam)
@littlefermatАй бұрын
Oh thanks for the info! and good luck to you
@bata3258Ай бұрын
discontinuing inequalities?
@caiodavi9829Ай бұрын
😢
@littlefermatАй бұрын
Nope but I was having fun with this bmo1 paper so I decided to post my solutions. I will surely continue the inequalities series once this is uploaded
@bata3258Ай бұрын
@@littlefermat alr, do what u think is fun
@msdmathssousdopamine8630Ай бұрын
Great video. In fact, the identities are valid for x different from 0 and 1. You can check that for all real numbers a, the piecewise function defined by f(x) = {x if x<>0 / a if x=0 / a+1 if x=1 satisfies the functional equation.
@hellomellow-jd5qvАй бұрын
Thanks for the awesome video! I always love your content. Do you happen to offer any online coaching for the IMO? Love from Syria 🤍
@GWHARSHAN2 ай бұрын
How do you write pie -2x , plz explain
@Jididududud2 ай бұрын
Next inequality? Cauchy?
@littlefermat2 ай бұрын
Yep
@Cooososoo2 ай бұрын
Thnx❤
@SwastikaDey-gq4ex2 ай бұрын
we deserve a projective geometry course bro. give that as an early Christmas gift.
@expSenpai2 ай бұрын
Thanks bro
@torung96062 ай бұрын
thank you so much <3
@asmadaamouche14672 ай бұрын
Hi can you contact with me an important matter?
@littlefermat2 ай бұрын
You can reach out to me on email at description
@Cooososoo2 ай бұрын
Is functional equations your favourite?
@littlefermat2 ай бұрын
In algebra definitely, but I like inequalities too!